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This video is going to be about linear inequality word problems. So let's
look at the first problem.
Mary has $33 to buy pizza and soda for a party.
Pizzas cost $12.50 each
and she needs to buy two pizzas.
If a large bottle of soda costs $1.50,
how many bottles can she buy without going over her $33 budget?
how many bottles can she buy without going over her $33 budget?
Okay, so she's got a total of $33.
She has to buy two pizzas,
so
we know that the cost for the pizza
is going to be 2
times $12.50.
And she wants to know how many bottles of soda she can buy. So the number of bottles
of soda will just be x.
And each bottle of soda will cost her
$1.50. So the cost of the soda is going to be x times $1.50.
$1.50. So the cost of the soda is going to be x times $1.50.
And that's got to deal with less than or equal to
$33.00.
She can spend less than $33,
or she can spend exactly $33.
So let's see what happens.
Well 2 times $12.50 is $25.
And I've got x times
$1.50.
I'm just gonna write that as 1.5x,
I'm just gonna write that as 1.5x,
is less than or equal to
33.
So I'll subtract 25 from both sides.
And now I've got 1.5x
is less than or equal to ... let's see 33 minus 25 is 8.
And I've got to divide by 1.5.
So
x
equals 8
over 1.5. Now here's a nice trick:
instead of trying to reduce this fraction, or divide, I'm going to
multiply
the numerator and the denominator by 2. That way i won't have a decimal
in the denominator.
So if I multiply the numerator by 2,
I'll have 16.
If I multiply the denominator by 2, I've got 3.
And then...
oh, I'm sorry. This should be less than or equal to. I made an
"equal to" sign.
And
16 divided by 3 is
5
and 1/3. But you can't buy a third of a bottle of soda,
so the most soda she can buy
will be five bottles. She'll have some money left over, but there's nothing she can do with
it.
It's not enough to buy another bottle.
So all we did in this problem
was
look at the amount of money
that she must spend. She must spend less than 2 times 12.50.
And then
look at how much more she might spend. That would be x times
$1.50.
We don't know what x is yet, and we know that that has to be less than or equal
to the total amount, 33. So now we've just got a linear inequality.
We just multiply 2 times the $12.50 and so on
and come down to our answer.
Remember
the fraction part of the answer
is not going to help. So the answer to this is
she can buy five bottles of soda.
Okay, let's look at one more.
Harry has taken 4 test in his chemistry class
and has gotten the following grades:
93,
83, 94,
and 88.
He has to take one more test,
and he will get an 'A' for the course if his average for all the tests
is at least 90.
What is the lowest grade he can get on the fifth test
and still get an 'A' for the course?
So remember,
the average is going to be
all of the test scores
added together
divided by the number of tests. So we want to add together
93
plus 83
plus 94
plus 88
plus... We don't know what the last test is...
plus x.
That's going to be five tests,
so we'll have to divide that by 5.
And that has to be
at least 90.
So that's got to be greater than or equal to
90.
Okay, I've added these up in advance and these 4 numbers add up to 358.
Okay, I've added these up in advance and these 4 numbers add up to 358.
So I have 358
plus x
over 5
is greater than or equal to
90.
Well the next step is going to be to multiply by 5.
So 358
plus x... that's what's gonna happen when I multiply this side by 5.
Multiplying the right side by 5...
5 times 9 is 45, so 5 times 90 would be 450.
5 times 9 is 45, so 5 times 90 would be 450.
And then,
I'll subtract 358 from both sides.
And I have x is
greater than or equal to...
let's see, that a 2 and a 9...
x has to be greater than or equal to
92.
In other words, if he gets a 92 on his final test or better,
then he'll get an 'A' for the course.
And we could check this:
When we add up all of these numbers, 93 plus 83 plus 94 plus 88
plus x, instead of x we'll put in 92.
And we'll make sure that when we add these together and divide by 5,
it's at least as big as 90.
Okay, so
that's about it.
Take care, I'll see you next time.